Exercise 7.5

a)

The square of a diagonal matrix is easy to compute, just square the diagonal elements. According to the rules of matrix multiplication we get: \[ \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix}^2 = \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix} \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix} = \begin{pmatrix} a^2 & 0 \\ 0 & b^2 \end{pmatrix} \]

b)

\[ \rho = \begin{pmatrix} \frac{1}{3} & 0 \\ 0 & \frac{2}{3} \end{pmatrix} \]

\[ \rho^2 = \begin{pmatrix} \frac{1}{9} & 0 \\ 0 & \frac{4}{9} \end{pmatrix} \]

\[ \mathrm{Tr} (\rho ) = \frac{1}{3} + \frac{2}{3} = 1 \]

\[ \mathrm{Tr} (\rho^2 ) = \frac{1}{9} + \frac{4}{9} = \frac{5}{9} < 1 \]

c)

If \(\rho\) is a density matrix, it represents a mixed state, since \(\rho^2 \neq \rho\) or \(\mathrm{Tr} (\rho^2) < 1\).